January 20, 2014

Two polymath (of a sort) proposed projects

This post is meant to propose and discuss a polymath project and a sort of polymath project.

I. A polymath proposal: Convex hulls of real algebraic varieties.

One of the interesting questions regarding the polymath endeavor was:

Can polymath be used to develop a theory/new area?

My idea is to have a project devoted to develop a theory of “convex hulls of real algebraic varieties”. The case where the varieties are simply a finite set of points is a well-developed area of mathematics – the theory of convex polytopes, but the general case was not studied much. I suppose that for such a project the first discussions will be devoted to raise questions/research directions. (And mention some works already done.)

In general (but perhaps more so for an open-ended project), I would like to see also polymath projects which are on longer time scale than existing ones but perhaps less intensive, and that people can “get in” or “spin-off” at will in various times.

II. A polymath-of-a-sort proposal: Statements about the Riemann Hypothesis

The Riemann hypothesis is arguably the most famous open question in mathematics. My view is that it is premature to try to attack the RH by a polymath project (but I am not an expert and, in any case, a project of this kind is better conducted with some specific program in mind). I propose something different. In a sort of polymath spirit the project I propose invite participants, especially professional mathematicians who thought about the RH over the years, to share their thoughts about RH.

Ideally each comment will be

1) One or a few paragraphs long

2) Well-thought, focused and rather polished

A few comments by the same contributors are also welcome.

To make it clear, the thread I propose is not going to be a research thread and also not a place for further discussions beyond some clarifying questions. Rather it is going to be a platform for interested mathematician to make statements and expressed polished thoughts about RH. (Also, if adopted, maybe we will need a special name for such a thing.)

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This thread is not launching any of the two suggested projects, but rather a place to discuss further these proposals. For the second project, it will be better still if the person who runs it will be an expert in the area, and certainly not an ignorant. For the first project, maybe there are better ideas for areas/theories appropriate for polymathing.

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For your first proposal, can you give some more details as to what you have in mind: some examples perhaps, some vague questions/conjectures, and pointers to existing work. I know some convex geometry and optimization but your proposal doesn’t immediately give me any ideas. Is this the kind of stuff Parillo and others do?

“some examples perhaps, some vague questions/conjectures, and pointers to existing work” My idea is that the first post will be devoted to discussing it. Given that there is so much to say about convex hulls of finite sets of points, I suppose that there are many questions that can be asked about convex hulls of other real semi algebraic varieties. Curves, for example.

like the riemann hypothesis idea lots. a “thinking outside the box” use of polymath. actually have been thinking lately it could be really interesting to use a stackexchange chatroom to discuss a particular famous problem.
“I propose invite participants, especially professional mathematicians who thought about the RH over the years, to share their thoughts about RH”. it seems this is saying the comments are by invitation only? how would you determine the list of invitees?

stackexchange is both powerful/restrictive at the same time… an unusual combination… hope that someday something with more flexibility arises, dont know what it would look like exactly…. to me the system also does have some significant cultural baggage….

Can we characterize which convex subset of the plane is the convex hull of a semialgebraic set? If this semialgebraic set is a collection of points, then it’s convex hull is a polygon, possibly degenerate.

About RH, one idea could be to consider field automorphisms of $\mathbb{C}$ that commute with $\zeta$ and proving such automorphisms are continuous. Maybe this could help prove the global invariance of the set of non-trivial zeroes of $\zeta$ under such automorphisms, that thus would be isometries. In other words: formalize the fact that “preserving” $\zeta$ boils down to “preserving its set of non-trivial zeroes”.

More precisely, given a field automorphism of $\mathbb{C}$, that we will denote by $\sigma$, that commutes with $\zeta$, let’s consider the operator $\phi_{\sigma}:F\mapsto \sigma\circ F\sigma^{-1}$. Such an operator will be called an automorphism of $\zeta$, since one has $\zeta=\phi_{\sigma}(\zeta)$. Moreover, one can build a group morphism from the group of isometries of the complex plane that preserve globally the multiset of non-trivial zeroes of $\zeta$ to the group of automorphisms of $\zeta$. 2 things remain to be done: 1) proving that $\sigma$ is continuous 2) deducing from that that the considered group morphism is actually an isomorphism.